Whakaoti i te {r}{22x+3y=5}{3x+2y=70}

$$\left. \begin{array} { r } { 22 x + 3 y = 5 } \\ { 3 x + 2 y = 70 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{40}{7} = -5\frac{5}{7} \approx -5.714285714$$

$$y = \frac{305}{7} = 43\frac{4}{7} \approx 43.571428571$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$22x+3y=5,3x+2y=70$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$22x+3y=5$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$22x=-3y+5$$

Whakawehea ngā taha e rua ki te 22.

$$x=\frac{1}{22}\left(-3y+5\right)$$

Whakareatia \frac{1}{22} ki te -3y+5.

$$x=-\frac{3}{22}y+\frac{5}{22}$$

Whakakapia te \frac{-3y+5}{22} mō te x ki tērā atu whārite, 3x+2y=70.

$$3\left(-\frac{3}{22}y+\frac{5}{22}\right)+2y=70$$

Whakareatia 3 ki te \frac{-3y+5}{22}.

$$-\frac{9}{22}y+\frac{15}{22}+2y=70$$

Tāpiri -\frac{9y}{22} ki te 2y.

$$\frac{35}{22}y+\frac{15}{22}=70$$

Me tango \frac{15}{22} mai i ngā taha e rua o te whārite.

$$\frac{35}{22}y=\frac{1525}{22}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{35}{22}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{305}{7}$$

Whakaurua te \frac{305}{7} mō y ki x=-\frac{3}{22}y+\frac{5}{22}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{22}\times \frac{305}{7}+\frac{5}{22}$$

Whakareatia -\frac{3}{22} ki te \frac{305}{7} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{915}{154}+\frac{5}{22}$$

Tāpiri \frac{5}{22} ki te -\frac{915}{154} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{40}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{40}{7},y=\frac{305}{7}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$22x+3y=5,3x+2y=70$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}22&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\70\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}22&3\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}22&3\\3&2\end{matrix}\right))\left(\begin{matrix}22&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}22&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}22&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}22&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{22\times 2-3\times 3}&-\frac{3}{22\times 2-3\times 3}\\-\frac{3}{22\times 2-3\times 3}&\frac{22}{22\times 2-3\times 3}\end{matrix}\right)\left(\begin{matrix}5\\70\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{35}&-\frac{3}{35}\\-\frac{3}{35}&\frac{22}{35}\end{matrix}\right)\left(\begin{matrix}5\\70\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{35}\times 5-\frac{3}{35}\times 70\\-\frac{3}{35}\times 5+\frac{22}{35}\times 70\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{40}{7}\\\frac{305}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{40}{7},y=\frac{305}{7}$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$22x+3y=5,3x+2y=70$$

Kia ōrite ai a 22x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 22.

$$3\times 22x+3\times 3y=3\times 5,22\times 3x+22\times 2y=22\times 70$$

Whakarūnātia.

$$66x+9y=15,66x+44y=1540$$

Me tango 66x+44y=1540 mai i 66x+9y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$66x-66x+9y-44y=15-1540$$

Tāpiri 66x ki te -66x. Ka whakakore atu ngā kupu 66x me -66x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y-44y=15-1540$$

Tāpiri 9y ki te -44y.

$$-35y=15-1540$$

Tāpiri 15 ki te -1540.

$$-35y=-1525$$

Whakawehea ngā taha e rua ki te -35.

$$y=\frac{305}{7}$$

Whakaurua te \frac{305}{7} mō y ki 3x+2y=70. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times \frac{305}{7}=70$$

Whakareatia 2 ki te \frac{305}{7}.

$$3x+\frac{610}{7}=70$$

Me tango \frac{610}{7} mai i ngā taha e rua o te whārite.

$$3x=-\frac{120}{7}$$

Whakawehea ngā taha e rua ki te 3.

$$x=-\frac{40}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{40}{7},y=\frac{305}{7}$$